Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(2,3)(6,7)(9,11,15,20)(10,16,19,24)(12,23,17,14)(13,18,22,21)(27,31)(28,34)(35,37)(36,39);(1,3,5,2)(4,7,8,6)(11,20)(14,23)(16,24)(18,21)(25,27,30,31)(26,28,33,34)(29,37,38,35)(32,39,40,36);(1,4,5,8)(2,6,3,7)(9,12)(10,13)(11,14)(15,17)(16,21)(18,24)(19,22)(20,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,13)(10,17)(11,21)(12,19)(14,24)(15,22)(16,23)(18,20)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(9,14)(10,18)(11,12)(13,24)(15,23)(16,22)(17,20)(19,21)(25,29)(26,32)(27,35)(28,36)(30,38)(31,37)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313858);